In this first practical we are going to fit spatial modes for Areal, Geostatistical and Point-Process Data.
Areal Data
Models for areal data
In this practical we are going to fit an areal model. We will:
Learn how to fit an areal model in inlabru
Learn how to add spatial covariates to the model
Learn how to do predictions
Learn how to simulate from the fitted model
Libraries to load:
library(dplyr)library(INLA)library(ggplot2)library(patchwork)library(inlabru) library(mapview)# load some libraries to generate nice map plotslibrary(scico)
The data
We consider data on respiratory hospitalizations for Greater Glasgow and Clyde in 2007. The data are available from the CARBayesdata R Package:
The pollutionhealthdata contains the spatiotemporal data on respiratory hospitalizations, air pollution concentrations and socio-economic deprivation covariates for the 271 Intermediate Zones (IZ) that make up the Greater Glasgow and Clyde health board in Scotland. Data are provided by the Scottish Government and the available variables are:
IZ: unique identifier for each IZ.
year: the year when the measurements were taken
observed: observed numbers of hospitalizations due to respiratory disease.
expected: expected numbers of hospitalizations due to respiratory disease computed using indirect standardisation from Scotland-wide respiratory hospitalization rates.
pm10: Average particulate matter (less than 10 microns) concentrations.
jsa: The percentage of working age people who are in receipt of Job Seekers Allowance
price: Average property price (divided by 100,000).
The GGHB.IZ data is a Simple Features (sf) object containing the spatial polygon information for the set of 271 Intermediate Zones (IZ), that make up of the Greater Glasgow and Clyde health board in Scotland ( Figure 1 ).
Figure 1: Greater Glasgow and Clyde health board represented by 271 Intermediate Zones
We first merge the two dataset and select only one year of data, compute the SME and plot the observed
Then we compute the adjacency matrix using the functions poly2nb() and nb2mat() in the spdep library. We then convert the adjacency matrix into the precision matrix \(\mathbf{Q}\) of the CAR model. Remember this matrix has, on the diagonal the number of e
cmp =~Intercept(1) +space(space, model ="besag", graph = Q) +iid(space, model ="iid")formula = observed ~ Intercept + space + iidlik =bru_obs(formula = formula, family ="poisson",E = expected,data = resp_cases)fit =bru(cmp, lik)
After fitting the model we want to extract results.
Tip Question
What is the estimated value for \(\beta_0\)?
Look at the estimated values of the hyperparameters using fit$summary.hyperpar , which of the two spatial components (structured or unstructured) explains more of the variability in the counts?
We now look at the predictions over space.
Warning Task
Complete the code below to produce prediction of the linear predictor \(\eta_i\) and of the risk \(\lambda_i\) and of the expected cases \(E_i\exp(\lambda_i)\) over the whole space of interest. Then plot the mean and sd of the resulting surfaces.
Note: Here we are predicting the mean of counts, not the counts!!! Predicting counts is the theme of the next task!
Getting prediction densities
Posterior predictive distributions, that is \(\pi(y_i^{\text{new}}|\mathbf{y})\) are of interest in many applied problems. The bru() function does not return predictive densities. In the previous step we have computed predictions for the expected counts\(\pi(E_i\lambda_i|\mathbf{y})\). The predictive distribution is then: \[
\pi(y_i^{\text{new}}|\mathbf{y}) = \int \pi(y_i|E_i\lambda_i)\pi(E_i\lambda_i|\mathbf{y})\ dE_i\lambda_i
\] where, in our case, \(\pi(y_i|E_i\lambda_i)\) is Poisson with mean \(E_i\lambda_i\). We can achieve this using the following algorith:
Simulate \(n\) replicates of \(g^k = E_i\lambda_i\) for \(k = 1,\dots,n\) using the function generate() which takes the same input as predict()
For each of the \(k\) replicates simulate a new value \(y_i^{new}\) using the function rpois()
Summarise the \(n\) samples of \(y_i^{new}\) using, for example the mean and the 0.025 and 0.975 quantiles.
Here is the code:
# simulate 1000 realizations of E_i\lambda_iexpected_counts =generate(fit, resp_cases, ~ expected *exp(Intercept + space),n.samples =1000)# simulate poisson dataaa =rpois(271*1000, lambda =as.vector(expected_counts))sim_counts =matrix(aa, 271, 1000)# summarise the samples with posterior means and quantilespred_counts =data.frame(observed = resp_cases$observed,m =apply(sim_counts,1,mean),q1 =apply(sim_counts,1,quantile, 0.025),q2 =apply(sim_counts,1,quantile, 0.975),vv =apply(sim_counts,1,var) )
Warning Task
Plot the observations against the predicted new counts and the predicted expected counts. Include the uncertainty and compare the two.
Code
ggplot() +geom_point(data = pred_counts, aes(observed, m, color ="Pred_obs")) +geom_errorbar(data = pred_counts, aes(observed, ymin = q1, ymax = q2, color ="Pred_obs")) +geom_point(data = pred$cases, aes(observed, mean, color ="Pred_means")) +geom_errorbar(data = pred$cases, aes(observed, ymin = q0.025, ymax = q0.975, color ="Pred_means")) +geom_abline(intercept =0, slope =1)
Adding a spatial covariate
Finally, we want to add a covariate to the model. We are going to check if the PM10 levels influence the numbers of hospitalizations.
Warning Task
Add the effect of PM10 in the previous model. First as a linear effect and then as a smooth effect (RW2). Check the results.
To use the RW2 model you first need to group the values of the PM10 using the inla.group() function.
Code
## Linear Effectcmp =~Intercept(1) +space(space, model ="besag", graph = Q) +iid(space, model ="iid") +cov(pm10, model ="linear")formula = observed ~ Intercept + space + iid + covlik =bru_obs(formula = formula, family ="poisson",E = expected,data = resp_cases)fit_lin =bru(cmp, lik)# effect of the covaraite# fit_lin$summary.fixed## Linear Effectresp_cases$pm10_group=inla.group(resp_cases$pm10)cmp =~Intercept(1) +space(space, model ="besag", graph = Q) +iid(space, model ="iid") +cov(pm10_group, model ="rw2")formula = observed ~ Intercept + space + iid + covlik =bru_obs(formula = formula, family ="poisson",E = expected,data = resp_cases)fit_smooth =bru(cmp, lik)#check the smooth effect of the covariate#fit_smooth$summary.random$cov %>% ggplot() + geom_line(aes(ID,mean)) +# geom_ribbon(aes(ID, ymin =`0.025quant`, ymax = `0.975quant`), alpha = 0.5)
Geostatistics
Geostatistics
In this practical we are going to fit a geostatistical model. We will:
Learn how to fit a geostatistical model in inlabru
Learn how to build a mesh for the SPDE representation
Learn how to add spatial covariates to the model
Learn how to do predictions
Learn how to simulate from the fitted model
Libraries to load:
library(dplyr)library(INLA)library(inlabru) library(sf)library(terra)# load some libraries to generate nice map plotslibrary(scico)library(ggplot2)library(patchwork)library(mapview)library(tidyterra)
The data
In this practical, we will explore data on the Pacific Cod (Gadus macrocephalus) from a trawl survey in Queen Charlotte Sound. The pcod dataset is available from the sdmTMB package and contains the presence/absence records of the Pacific Cod during each surveys along with the biomass density of Pacific cod in the area swept (kg/Km\(^2\)). The qcs_grid data contain the depth values stored as \(2\times 2\) km grid for Queen Charlotte Sound.
The dataset contains presence/absence data from 2003 to 2017. In this practical we only consider year 2003.
We first load the dataset and select the year of interest
We convert the covariate into a raster and assign the same coordinate reference:
depth_r <-rast(qcs_grid, type ="xyz")crs(depth_r) <-crs(pcod_sf)
Finally we can plot our dataset. Note that to plot the raster we need to upload also the tidyterra library.
ggplot()+geom_spatraster(data=depth_r$depth)+geom_sf(data=pcod_sf,aes(color=factor(present))) +scale_color_manual(name="Occupancy status for the Pacific Cod",values =c("black","orange"),labels=c("Absence","Presence"))+scale_fill_scico(name ="Depth",palette ="nuuk",na.value ="transparent" ) +xlab("") +ylab("")
The model
We first fit a simple model where we consider the observation as Bernoulli and where the linear predictor contains only one intercept and the GR field defined through the SPDE approach. The model is defined as:
Stage 1 Model for the response
\[
y(s)|\eta(s)\sim\text{Binom}(1, p(s))
\]Stage 2 Latent field model
\[
\omega(s)\sim \text{ GF with range } \rho\ \text{ and maginal variance }\ \sigma^2
\]
Stage 3 Hyperparameters
The hyperparameters of the model are \(\rho\) and \(\sigma\)
NOTE In this case the linear predictor \(\eta\) consists of two components!!
The workflow
When fitting a geostatistical model we need to fulfill the following tasks:
Build the mesh
Define the SPDE representation of the spatial GF. This includes defining the priors for the range and sd of the spatial GF
Define the components of the linear predictor. This includes the spatial GF and all eventual covariates
Define the observation model using the bru_obs() function
Run the model using the bru() function
1. Building the mesh
The first task, when dealing with geostatistical models in inlabru is to build the mesh that covers the area of interest. For this purpose we use the function fm_messh_2d.
One way to build the mesh is to start from the locations where we have observations, these are contained in the dataset pcod_sf.
mesh =fm_mesh_2d(loc = pcod_sf, # Build the meshmax.edge =c(10,20), # The largest allowed triangle edge length.offset =c(5,50)) # The automatic extension distanceggplot() +gg(mesh) +geom_sf(data= pcod_sf, aes(color =factor(present)), size =0.1) +xlab("") +ylab("")
As you can see from the plot above, some of the locations are very close to each other, this causes some very small triangles. This can be avoided using the option cutoff = which collapses the locations that are closer than a cutoff (those points are collapsed in the mesh construction but, of course, not when it come to estimaation.)
mesh =fm_mesh_2d(loc = pcod_sf, # Build the meshcutoff =2,max.edge =c(10,20), # The largest allowed triangle edge length.offset =c(5,50)) # The automatic extension distanceggplot() +gg(mesh) +geom_sf(data= pcod_sf, aes(color =factor(present)), size =0.1) +xlab("") +ylab("")
Key parameters in mesh construction include: max.edge for maximum triangle edge lengths, offset for inner and outer extensions (to prevent edge effects), and cutoff to avoid overly small triangles in clustered areas.
Note
General guidelines for creating the mesh
Create triangulation meshes with fm_mesh_2d()
Move undesired boundary effects away from the domain of interest by extending to a smooth external boundary
Use a coarser resolution in the extension to reduce computational cost (max.edge=c(inner, outer))
Use a fine resolution (subject to available computational resources) for the domain of interest (inner correlation range) and filter out small input point clusters (0 < cutoff < inner)
Coastlines and similar can be added to the domain specification in fm_mesh_2d() through the boundary argument.
Warning Task
Look at the documentation for the fm_mesh_2d function typing
?fm_mesh_2d
play around with the different options and create different meshes.
The rule of thumb is that your mesh should be:
fine enough to well represent the spatial variability of your process, but not too fine in order to avoid computation burden
the triangles should be regular, avoid long and thin triangles.
The mesh should contain a buffer around your area of interest (this is what is defined in the offset option) in order to avoid boundary artefact in the estimated variance.
2. Define the SPDE representation of the spatial GF
To define the SPDE representation of the spatial GF we use the function inla.spde2.pcmatern. This takes as input the mesh we have defined and the PC-priors definition for \(\rho\) and \(\sigma\) (the range and the marginal standard deviation of the field).
PC priors Gaussian Random field are defined in (Fuglstad et al. 2018). From a practical perspective for the range \(\rho\) you need to define two paramters \(\rho_0\) and \(p_{\rho}\) such that you believe it is reasonable that
\[
P(\rho<\rho_0)=p_{\rho}
\]
while for the margianal variance \(\sigma\) you need to define two parameters \(\sigma_0\) and \(p_{\sigma}\) such that you believe it is reasonable that
\[
P(\sigma<\sigma_0)=p_{\sigma}
\]
You can use the following function to compute and plot the prior distributions for the range and sd of the Matern field.
dens_prior_range =function(rho_0, p_alpha){# compute the density of the PC prior for the# range rho of the Matern field# rho_0 and p_alpha are defined such that# P(rho<rho_0) = p_alpha rho =seq(0, rho_0*10, length.out =100) alpha1_tilde =-log(p_alpha) * rho_0 dens_rho = alpha1_tilde / rho^2*exp(-alpha1_tilde / rho)return(data.frame(x = rho, y = dens_rho))}dens_prior_sd =function(sigma_0, p_sigma){# compute the density of the PC prior for the# sd sigma of the Matern field# sigma_0 and p_sigma are defined such that# P(sigma>sigma_0) = p_sigma sigma =seq(0, sigma_0*10, length.out =100) alpha2_tilde =-log(p_sigma)/sigma_0 dens_sigma = alpha2_tilde*exp(-alpha2_tilde * sigma) return(data.frame(x = sigma, y = dens_sigma))}
Here are some alternatives for defining priors for our model
And here we plot the different priors for the range:
ggplot() +geom_line(data =dens_prior_range(30,.5), aes(x,y, color ="model1")) +geom_line(data =dens_prior_range(1000,.5), aes(x,y, color ="model2")) +geom_line(data =dens_prior_range(100,.5), aes(x,y, color ="model3"))
and for the sd:
ggplot() +geom_line(data =dens_prior_sd(1,.5), aes(x,y, color ="model1")) +geom_line(data =dens_prior_sd(10,.5), aes(x,y, color ="model2")) +geom_line(data =dens_prior_sd(.1,.5), aes(x,y, color ="model3"))
Tip Question
Consider the pcod_sf, the spatial extension and type of the data…is some of the previous choices more reasonable than other?
NOTE Remember that a prior should be reasonable..but the model should not totally depend on it.
3. Define the components of the linear predictor
We have now defined a mesh and a SPDE representation of the spatial GF. We now need to define the model components:
cmp =~Intercept(1) +space(geometry, model = spde_model3)
NOTE since the dataframe we use (pcod_sf) is an sf object the input in the space() component is the geometry of the dataset.
4. Define the observation model
Our data are Bernoulli distributed so we can define the observation model as:
formula = present ~ Intercept + spacelik =bru_obs(formula = formula, data = pcod_sf, family ="binomial")
5. Run the model
Finally we are ready to run the model
fit1 =bru(cmp,lik)
Extract results
Hyperparameters
Warning Task
Plot the posterior densities for the range \(\rho\) and the standard deviation \(\sigma\) alog with the prior for both parameters.
Posterior marginals for the hyperparameters can be extracted from the fitted model as:
Code
fit1$marginals.hyperpar$'name of the parameter'
Code
# Extract marginal for the rangeggplot() +geom_line(data = fit1$marginals.hyperpar$`Range for space`,aes(x,y, color ="Posterior")) +geom_line(data =dens_prior_range(100,.5),aes(x,y, color ="Prior"))ggplot() +geom_line(data = fit1$marginals.hyperpar$`Stdev for space`,aes(x,y, color ="Posterior")) +geom_line(data =dens_prior_sd(1,.5), aes(x,y, color ="Prior"))
Spatial prediction
We now want to extract the estimated posterior mean and sd of spatial GF. To do this we first need to define a grid of points where we want to predict. We do this using the function fm_pixel() which creates a regular grid of points covering the mesh
pxl =fm_pixels(mesh)
then compute the prediction for both the spatial GF and the linear predictor (spatial GF + intercept)
Note The posterior sd is lowest at the observation points. Note how the posterior sd is inflated around the border, this is the “border effect” due to the SPDE representation.
Instead of predicting over a grid covering the whole mesh, we can limit our predictions to the points where the covariate is defined. We can do this by defining a sf object using coordinates in the object depth_r.
Instead of computing the posterior mean and standard deviations of the estimated surface, one can also simulate possible realizations of such surface. This will give the user a better idea of the type of realized surfaces one can expect. We can do this using the function generate().
# we simulate 4 samples from the gens =generate(fit1, pxl1, ~ (Intercept + space),n.samples =4)pp =cbind(pxl1, gens)pp %>%select(-depth) %>%pivot_longer(-geometry) %>%ggplot() +geom_sf(aes(color = value)) +facet_wrap(.~name) +scale_color_scico(direction =-1) +ggtitle("Sample from the fitted model")
An alternative model
We now want to check if the depth covatiate has an influende on the probability of presence. We do this in two different models
Model 1 The depth enters the model in a linear way. The linear predictor is then defined as:
Model 1 The depth enters the model in a non linear way. The linear predictor is then defined as:
\[
\eta(s) = \text{logit}(p(s)) = \beta_0 + \omega(s) + f(\text{depth}(s))
\] where \(f(.)\) is a smooth function. We will use a RW2 model for this.
Warning Task
Fit model 1. Define components, observation model and use the bru() function to estimate the parameters.
Note Use the scaled version of the covariate stored in depth_r$depth_scaled.
What is the liner effect of depth on the logit probability?
Add hint details here…
Code
cmp =~Intercept(1) +space(geometry, model = spde_model3) +covariate(depth_r$depth_scaled, model ="linear")formula = present ~ Intercept + space + covariatelik =bru_obs(formula = formula, data = pcod_sf, family ="binomial")fit2 =bru(cmp, lik)
We now want to fit Model 2 where we allow the effect of depth to be non-linear. To use the RW2 model we need to group the values of depth into distinct classe. To do this we use the function inla.group() which, by default, creates 20 groups. The we can fit the model as usual
# create the grouped variabledepth_r$depth_group =inla.group(values(depth_r$depth_scaled))# run the modelcmp =~Intercept(1) +space(geometry, model = spde_model3) +covariate(depth_r$depth_group, model ="rw2")formula = present ~ Intercept + space + covariatelik =bru_obs(formula = formula, data = pcod_sf, family ="binomial")fit3 =bru(cmp, lik)# plot the estimated effect of depthfit3$summary.random$covariate %>%ggplot() +geom_line(aes(ID,mean)) +geom_ribbon(aes(ID, ymin =`0.025quant`, ymax =`0.975quant`), alpha =0.5)
Warning Task
Create a map of predicted probability from Model 3. You can use a inverse logit function defined as
inv_logit =function(x) (1+exp(-x))^(-1)
The predict() function can take as input also functions of elements of the components you want to consider
Code
inv_logit =function(x) (1+exp(-x))^(-1)pred3 =predict(fit3, pxl1, ~inv_logit(Intercept + space + covariate) )pred3 %>%ggplot() +geom_sf(aes(color = mean)) +scale_color_scico(direction =-1) +ggtitle("Sample from the fitted model")
In this practical we consider the data clmfires in the spatstat library.
This dataset is a record of forest fires in the Castilla-La Mancha region of Spain between 1998 and 2007. This region is approximately 400 by 400 kilometres. The coordinates are recorded in kilometres. For more info about the data you can type:
?clmfires
We first read the data and transform them into an sf object. We also create a polygon that represents the border of the Castilla-La Mancha region. We select the data for year 2004 and only those fires caused by lightning.
Figure 2: Distribution of the observed forest fires caused by lightning in Castilla-La Mancha in 2004
Fit a homogeneous Poisson Process
As a first exercise we are going to fit a homogeneous Poisson process (HPP) to the data. This is a model that assume constant intensity over the whole space so our linear predictor is then:
Where \(N_k\) is the number of integration points \(s_1,\dots,s_{N_k}\) and \(w_1,\dots,w_{N_k}\) are the integration weights.
In this case, since the intensity is constant, the integration scheme is really simple: it is enough to consider one random point inside the domain with weight equal to the area of the domain.
# define integration schemeips =st_sf(geometry =st_sample(region, 1)) # some random location inside the domainips$weight =st_area(region) # integration weight is the area of the domaincmp =~0+beta_0(1)formula = geometry ~ beta_0lik =bru_obs(data = pp,family ="cp",formula = formula,ips = ips)fit1 =bru(cmp, lik)
Warning Task
Plot the estimated posterior distribution of the intensity
Compare the estimated expected number of fires on the whole domain with the observed ones.
Remember that in the inlabru framework we model the log intensity \(\eta = log(\lambda)\)
Code
# 1) The estimated posterior distribution of the intensity ispost_int =inla.tmarginal(function(x) exp(x), fit1$marginals.fixed$beta_0)post_int %>%ggplot() +geom_line(aes(x,y))
Code
# 2) To compute the expected number of points in the area we need to multiply the# estimated intensity by the area of the domain.# In the same plot we also show the number of observed fires as a vertical line.post_int =inla.tmarginal(function(x) st_area(region)*exp(x), fit1$marginals.fixed$beta_0)post_int %>%ggplot() +geom_line(aes(x,y)) +geom_vline(xintercept =dim(pp)[1])
Fit an Inhomogeneous Poisson Process
The model above has the clear disadvantages that assumes a constant intensity and from Figure 2 we clearly see that this is not the case.
The library spatstat contains also some covariates that can help explain the fires distribution. Figure @fit-altitude shows the location of fires together with the (scaled) altitude.
#|label: fig-altitude#|fig-cap: "Distribution of the observed forest fires and scaled altitude"#| elev_raster =rast(clmfires.extra[[2]]$elevation)elev_raster =scale(elev_raster)ggplot() +geom_spatraster(data = elev_raster) +geom_sf(data = pp) +scale_fill_scico()
We are now going to use the altitude as a covariate to explain the variability of the intensity \(\lambda(s)\) over the domain of interest.
Our model is \[
\log\lambda(s) = \beta_0 + \beta_1x(s)
\] where \(x(s)\) is the altitude at location \(s\).
Now we need to choose an integration scheme to solve the integral.
In this case we will take a simple grid based approach where each quadrature location has an equal weight. Our grid consists of \(N_k = 1000\) points and the weights are all equal to \(|\Omega|/N_k\).
Re-run the model changing the integration scheme in the ips input of the bru_obs() function.
Code
lik_bis =bru_obs(data = pp,family ="cp",formula = formula,ips = ips2)fit2bis =bru(cmp, lik_bis)# you can the check the differences between the two modelsrbind(fit2$summary.fixed,fit2bis$summary.fixed)
preds2 =predict(fit2, est_grid, ~data.frame(log_scale = Intercept + elev,lin_scale =exp(Intercept + elev)))# then visualize it likepreds2$log_scale %>%ggplot() +geom_sf(aes(color = mean)) +scale_color_scico()
Finally, we want to use the fitted model to estimate the total number of fires over the whole region. To do this we first have to fine the expected number of fires as:
where \(\beta_0\) is the intercept, \(\beta_1\) the effect of (standarized) altitude \(x(s)\) as before and \(u(s)\) is a Gaussian Random field defined through the SPDE approach.
Define the mesh
The first step, as any time we use the SPDE approach is to defie the mesh and the priors for the marginal variance and range:
We can then define the integration weight. Here we use the same points to define the SPDE approximation and to approximate the integral in the likelihood. We will see later that this does not have to be like this, BUT integration weight and SPDE weights have to be consistent with each other!
The model has three components: intercept, linear effect of altitude and the spatial GRF
Code
cmp =~Intercept(1) +space(geometry, model = spde_model) +elev(elev_raster, model ="linear")formula = geometry ~ Intercept + space + elevlik =bru_obs("cp",formula = formula,data = pp,ips = ips)fit3 =bru(cmp, lik)
Note when running the model above you will get a warning:
Warning in bru_log_warn(msg): Model input 'elev_raster' for 'elev' returned some NA values.
Attempting to fill in spatially by nearest available value.
To avoid this basic covariate imputation, supply complete data.
It means that the bru() function cannot find the covariate values for some of the mesh nodes. This is a common situation. As the warning says, the bru() function automatically imputes the value of the covarite using the nearest nodes. This increases the running time of the bru() function, so one solution is to impute the values of the covariate over the whole mesh ‘before’ running the bru() function.
Here, we notice that there is a single point for which elevation values are missing (see Figure 3 the red point that lies outside the raster extension ).
Figure 3: Integration scheme for numerical approximation of the stochastic integral in La Mancha Region
To solve this, we can increase the raster extension so it covers all both data-points and quadrature locations as well. Then, we can use the bru_fill_missing() function to input the missing values with the nearest-available-value. We can achieve this using the following code:
# Extend raster ext by 30 % of the original raster so it covers the whole meshre <-extend(elev_raster, ext(elev_raster)*1.3)# Convert to an sf spatial objectre_df <- re %>% stars::st_as_stars() %>%st_as_sf(na.rm=F)# fill in missing values using the original raster re_df$lyr.1<-bru_fill_missing(elev_raster,re_df,re_df$lyr.1)# rasterizeelev_rast_p <- stars::st_rasterize(re_df) %>%rast()ggplot() +geom_spatraster(data = elev_rast_p)
Note
The bru_fill_missing() function was added mainly to handle very local infilling on domain boundaries. For properly missing data, one should consider doing a proper model of the spatial field instead.
Results
Warning Task
Plot the estimated mean and standard deviation of the spatial GF and the log-intensity over the domain of interest
Instead of just looking at the posterior mean and standard deviation, it can be usefull to look at simulated fields from the posterior distribution. This is because the mean field is, by definition, smoother than any realization of the field. So looking at simulation can give us a better idea of how the field might look like. We can do this using the generate() function:
